Quantum Jacobi-Davidson Method

Computing electronic structures of quantum systems is a key task underpinning many applications in photonics, solid-state physics, and quantum technologies. This task is typically performed through iterative algorithms to find the energy eigenstates of a Hamiltonian, which are usually computationally expensive and suffer from convergence issues. In this work, we develop and implement the Quantum Jacobi-Davidson (QJD) method and its quantum diagonalization variant, the Sample-Based Quantum Jacobi-Davidson (SBQJD) method, and demonstrate their fast convergence for ground state energy estimation. We assess the intrinsic algorithmic performance of our methods through exact numerical simulations on a variety of quantum systems, including 8-qubit diagonally dominant matrices, 12-qubit one-dimensional Ising models, and a 10-qubit water molecule (H$_2$O) Hamiltonian. Our results show that both QJD and SBQJD achieve significantly faster convergence and require fewer Pauli measurements compared to the recently reported Quantum Davidson method, with SBQJD further benefiting from optimized reference state preparation. These findings establish the QJD framework as an efficient general-purpose subspace-based technique for solving quantum eigenvalue problems, providing a promising foundation for sparse Hamiltonian calculations on future fault-tolerant quantum hardware.

💡 Research Summary

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The paper introduces a new hybrid quantum‑classical algorithm called Quantum Jacobi‑Davidson (QJD) and its Sample‑Based variant (SBQJD) for efficiently solving eigenvalue problems that arise in electronic‑structure calculations. Building on the classical Jacobi‑Davidson (JD) method, which improves upon the Davidson algorithm by solving a correction equation that yields a search direction orthogonal to the current Ritz vector, the authors translate this framework to the quantum domain. The correction equation can be written in the form |t⟩ = A|rv⟩, where A is either the exact shifted‑inverse (H – E′I)⁻¹ multiplied by a scalar ε, or a preconditioned approximation εM⁻¹ – M⁻¹(H – E′I) when the Hamiltonian is diagonally dominant. Because A is generally non‑unitary, the authors employ the Linear Combination of Unitaries (LCU) technique to prepare the normalized correction state on a quantum processor. The LCU circuit uses ⌈log₂m⌉ ancilla qubits to encode the coefficients αi of the decomposition A = Σi αi Ui, and amplitude amplification reduces the overall cost from O(s²·polylog(s/p)) to O(s·polylog(s/p)), where s = Σi |αi| and p is the desired precision.

A key theoretical insight is that the correction step in QJD can be interpreted as a Newton‑type update for minimizing the Rayleigh quotient, which guarantees at least quadratic convergence of the eigenvalue error once the reference state has sufficient overlap with the true eigenstate. This contrasts with the linear convergence of first‑order methods such as the previously reported Quantum Davidson (QD) algorithm, which expands the subspace using the residual vector alone. Consequently, although each QJD iteration involves a more expensive inverse‑type operation, the total number of iterations required to reach chemical accuracy is dramatically reduced.

The authors also integrate QJD with Sample‑Based Quantum Diagonalization (SQDiag), a technique that samples the most frequently observed computational basis states from measurement outcomes to construct a reduced subspace V_SQDiag. By diagonalizing the Hamiltonian projected onto this subspace, SQDiag yields a compact basis that can serve as an efficient starting point for QJD. The resulting hybrid algorithm, SBQJD, benefits from a better initial reference and from the fact that the correction vector can be expressed using only the sampled basis, thereby lowering circuit depth and measurement overhead.

To benchmark the methods, the authors perform exact numerical simulations on three representative problems: (i) 8‑qubit diagonally dominant random matrices, (ii) a 12‑qubit one‑dimensional transverse‑field Ising model, and (iii) a 10‑qubit water (H₂O) Hamiltonian expressed in the STO‑3G minimal basis. For each case they compare QJD, SBQJD, and the baseline QD algorithm in terms of (a) number of subspace expansions (iterations) needed to achieve a target energy error, (b) total number of Pauli‑measurement shots, and (c) final energy error relative to the exact eigenvalue. The results show that QJD converges 3–5 times faster than QD, while SBQJD further reduces the required measurements by 20–40 % thanks to the optimized reference state. In the water molecule example, QJD reaches chemical accuracy (≤1 mHa) after 12 iterations, SBQJD after 9, whereas QD needs 35 iterations. Corresponding measurement counts are roughly 1.8 × 10⁶ (QJD), 1.2 × 10⁶ (SBQJD), and 5.6 × 10⁶ (QD) Pauli shots.

The paper includes a detailed resource analysis. While each QJD iteration requires an inverse‑type operation that is more demanding in terms of circuit depth and qubit count than the simple Hamiltonian‑application step of QD, the quadratic convergence dramatically reduces the overall runtime on fault‑tolerant hardware. The authors argue that on future error‑corrected quantum processors, the inverse operation can be implemented efficiently via Hamiltonian simulation combined with quantum phase estimation, making QJD a practical tool for high‑precision electronic‑structure calculations.

In the discussion, the authors note that the performance advantage of QJD is most pronounced for Hamiltonians that are either naturally diagonally dominant or for which an effective preconditioner M can be constructed. They also highlight that the method is flexible: alternative preconditioners, multi‑state extensions, and adaptive sampling strategies could further improve scalability. Potential extensions include integrating QJD with error mitigation techniques, exploring its behavior on noisy intermediate‑scale devices, and applying it to excited‑state problems or to larger molecular systems.

In conclusion, the Quantum Jacobi‑Davidson framework offers a substantial improvement over existing subspace‑based quantum eigensolvers. By leveraging Newton‑type updates via LCU‑based correction vectors and by coupling with sample‑based subspace construction, QJD and SBQJD achieve faster convergence and lower measurement overhead while retaining compatibility with both NISQ and fault‑tolerant quantum architectures. The work paves the way for more efficient quantum algorithms for electronic‑structure theory and other sparse‑matrix eigenvalue problems.